Meade got the job for two reasons I believe--
First, he hadn't explicitly said he wouldn't do it (as Reynolds apparently had).
Second, every officer approached, officially or unofficially, about it endorsed Meade. He had made the only really successful attack at Fredericksburg, and had been aggressive during the Chancellorsville Campaign. In particular, during Hooker's Council of War, Meade was the most forceful in favor of an attack.
Meade studiously stayed out of the infighting that developed after Chancellorsville. The two most prominent were Couch, who refused to serve any,longer under Hooker, and Slocum, who actively sought to undermine Hooker and have him relieved.